But you don't need to parameterize with k at all. In addition to x = y= 0, which is a solution, you have x != 0 and y != 0, which allows you to divide both sides by xy to get 1 = 1/x + 1/y. You can then rearrange to get, for instance, y = 1/(1-1/x) = x/(x-1). Since x=0, y=0 also works here, we don't need to keep it as an extra solution.